A085565 -id:A085565 - cp's OEIS frontend

A248589 Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.

1, 0, 6, 1, 8, 2, 4, 1, 3, 6, 4, 9, 0, 9, 6, 9, 6, 6, 2, 8, 0, 5, 3, 7, 8, 2, 8, 7, 3, 9, 8, 9, 4, 7, 1, 3, 1, 0, 0, 5, 5, 5, 9, 6, 4, 4, 7, 3, 2, 8, 8, 9, 2, 1, 2, 0, 4, 0, 5, 0, 1, 5, 1, 8, 3, 3, 8, 9, 8, 3, 3, 4, 5, 5, 6, 1, 2, 1, 1, 6, 1, 2, 4, 1, 3, 6, 9, 0, 0, 1, 0, 4, 2, 5, 9, 4, 5, 9, 0, 2

Offset: 1

Jean-François Alcover, Oct 09 2014

			1.061824136490969662805378287398947131005559644732889212...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 481.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's MathWorld, Square Point Picking

Cf. A053004, A085565, A096428, A096429.

RealDigits[Sqrt[2]*Pi^2/Gamma[1/4]^2, 10, 100][[1]]

sqrt(2)*Pi^2/gamma(1/4)^2 \\ G. C. Greubel, Jun 02 2017

I = sqrt(Pi/8)*(2-integral_{1..infinity} (sqrt(1+s^2)-s)*s^(-3/2) ds).
I = sqrt(Pi/2)*A053004, where A053004 is the arithmetic-geometric mean of 1 and sqrt(2).
I = Pi^(3/2)/(4*A085565), where A085565 is the lemniscate constant A.
I = sqrt(2)*Pi^2/Gamma(1/4)^2.

A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 7, 40, 45, 7, 308, 48, 975, 539, 88, 1664, 1105, 24255, 13376, 56576, 41769, 48279, 55936, 226304, 348075, 370139, 671232, 870400, 2082925, 4283037, 13872128, 80773120, 343682625, 4023459, 1553678336, 1900544, 14411758075, 59457783, 1471905792, 1406402560

Offset: 1

Devansh Singh, Aug 24 2020

a(n) is the numerator of (n/(n-1)) * ((n-3)/(n-2)) * ((n-4)/(n-5)) ...

			a(n)/A337355(n) equals 1, 2, 3/2, 2/3, 5/6, 9/5, 7/5, 40/63, 45/56, 7/4 ...
a(4) = numerator of (4*1)/(3*2) = numerator of 2/3 = 2.
a(5) = numerator of (5*2)/(4*3) = numerator of 5/6 = 5.
                      12  *   9*8  *  5*4  *  1
a(12) = numerator of --------------------------- = 48.
                        11*10  *  7*6  *  3*2

Cf. A337355 (denominators).

Cf. A076390, A085565, A007662.

a(n) = {numerator(prod(i=0, n-1, (n-i)^(-1)^((i+1)\2)))} \\ Andrew Howroyd, Aug 24 2020

a(n) = numerator of (n*A337355(n-2))/(a(n-2)*(n-1)) for n>=3.

Conjecture: a(4*n)/A337355(4*n) ~ 0.5990701173677... (=A076390). - Andrew Howroyd, Aug 25 2020

Terms a(31) and beyond from Andrew Howroyd, Aug 25 2020

A373534 Decimal expansion of Pi^(1/2)Gamma(1/20)/(10Gamma(11/20)).

Original entry on oeis.org

2, 1, 3, 5, 3, 4, 4, 9, 3, 3, 2, 4, 8, 0, 0, 4, 2, 2, 8, 0, 4, 6, 4, 7, 5, 2, 7, 9, 6, 8, 3, 7, 0, 6, 7, 7, 8, 8, 1, 0, 8, 7, 9, 3, 6, 6, 0, 1, 6, 4, 9, 4, 0, 0, 4, 0, 7, 7, 3, 1, 4, 4, 2, 9, 1, 0, 8, 7, 0, 3, 3, 0, 0, 1, 4, 9, 6, 8, 8, 3, 7, 8, 0, 6, 6, 5, 8, 3, 6, 5, 1, 2, 2, 2, 2, 2, 0, 5, 9, 6, 5

Offset: 1

Takayuki Tatekawa, Jun 08 2024

Constants from generalized Pi integrals: the case of n=20.

			2.135344933248004228046475279683...

Takayuki Tatekawa, Table of n, a(n) for n = 1..10001

Cf. A085565, A113477, A262427, A371824, A371881, A371930, A372111, A372327.

(2*sqrt(Pi)*GAMMA(21/20))/GAMMA(11/20): evalf(%, 102); # Peter Luschny, Jun 17 2024

RealDigits[2*Sqrt[Pi]/20*Gamma[1/20]/Gamma[11/20], 10, 5001][[1]]

Equals 2*Integral_{x=0..1} dx/sqrt(1-x^20).

Equals (2*sqrt(Pi)*Gamma(21/20))/Gamma(11/20). - Peter Luschny, Jun 17 2024

A371860 Decimal expansion of Integral_{x=0..1} 1 / sqrt(1 - x^3) dx.

Original entry on oeis.org

1, 4, 0, 2, 1, 8, 2, 1, 0, 5, 3, 2, 5, 4, 5, 4, 2, 6, 1, 1, 7, 5, 0, 1, 9, 0, 7, 9, 0, 5, 0, 2, 9, 4, 1, 3, 5, 4, 6, 3, 0, 2, 2, 2, 0, 5, 4, 2, 3, 9, 8, 6, 0, 9, 6, 1, 8, 1, 9, 9, 3, 9, 8, 7, 0, 7, 6, 2, 8, 4, 7, 6, 5, 9, 8, 1, 8, 0, 3, 2, 9, 6, 0, 7, 0, 8, 5, 2, 2, 6, 6, 4, 8, 5, 0, 2, 4, 7, 8, 4, 7, 0, 5, 5

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.4021821053254542611750190790502941354630222...

Decimal expansions of Integral_{x=0..1} 1 / sqrt(1 - x^k) dx: A019669 (k=2), this sequence (k=3), A085565 (k=4).

Cf. A002161, A118292, A202623, A203145.

RealDigits[Sqrt[Pi] Gamma[4/3]/Gamma[5/6], 10, 104][[1]]
RealDigits[Gamma[1/3]^3 / (2^(4/3)*Sqrt[3]*Pi), 10, 104][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

Equals sqrt(Pi) * Gamma(4/3) / Gamma(5/6).
Equals Gamma(1/3)^3 / (2^(4/3) * sqrt(3) * Pi). - Vaclav Kotesovec, Apr 09 2024
Equals A118292/2. - Hugo Pfoertner, Apr 09 2024

cp's OEIS Frontend

A248589 Decimal expansion of I, a constant appearing (as I^2) in the asymptotic variance of the area of the convex hull of random points in the unit square.

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A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).

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A373534 Decimal expansion of Pi^(1/2)*Gamma(1/20)/(10*Gamma(11/20)).

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A371860 Decimal expansion of Integral_{x=0..1} 1 / sqrt(1 - x^3) dx.

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A373534 Decimal expansion of Pi^(1/2)Gamma(1/20)/(10Gamma(11/20)).